powers of elliptic scator numbers

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Powers of elliptic scator numbers

Manuel Fernandez-Guasti 1

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Citation: Fernandez-Guasti M.

Powers of elliptic scator numbers.

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1 Lab. de Óptica Cuántica, Depto. de Física, Universidad Autónoma Metropolitana - Iztapalapa, C.P. 09340,Ciudad de México, A.P. 55-534, MEXICO.; [email protected]

Abstract: Elliptic scator algebra is possible in 1 + n dimensions, n ∈ N. It is isomorphic to complexalgebra in 1+1 dimensions, when the real part and any one hypercomplex component are considered.It is endowed with two representations: an additive one, where the scator components are representedas a sum; and a polar representation, where the scator components are represented as products ofexponentials. Within the scator framework, De Moivre’s formula is generalized to 1 + n dimensionsin the so called victoria equation. This novel formula is then used to obtain compact expressionsfor the integer powers of scator elements. A scator in S1+n can be factored into a product of nscators that are geometrically represented as its projections onto n two dimensional planes. Ageometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis isexpounded. The powers of scators, when the ratio of their director components is a rational number,lie on closed curves. For 1+2 dimensional scators, twisted curves in a three dimensional space areobtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in1+2 dimensions.

Keywords: Functions of hypercomplex variables; Algebraic geometry; Scator algebra.

MSC: 30G35; 13N15; 14C25

1. Introduction

De Moivre’s formula establishes a relationship between complex algebra and trigonom-etry through the evaluation of powers of numbers in modulus and angle variables. Thistype of relationship has been extended to higher dimensional algebras, notably quaternions[1], split quaternions [2], dual complex numbers [3], real and complex 2× 2 matrices [4] andother Clifford algebras [5]. De Moivre’s relationship is closely related to Euler’s formulaand hence to the exponential function. Advances in De Moivre type expressions havealso produced novel results in the exponential mapping, for example in split quaternionicmatrices [6]. The exponential function is unique, between other things, because for the realand complex algebras, it maps the additive group onto the multiplicative group, the basicoperations of the real and complex fields.

Elliptic scator algebra extends complex numbers to a higher number of dimensions,retaining one real part, labeled scalar part in scator parlance and any number of hyper-complex components [7]. The set of elliptic scators is akin to split-quaternions in the sensethat it contains zero divisors and nilpotent elements. However, the scator product is notbilinear, thus, it cannot be represented as a matrix - matrix product. The scator productprovides an interesting route for a unified mathematical description of quantum dynamics,encompassing the quantum system time evolution and its reduction to observed states [8].Scator algebra has also been successfully applied to other problems, such as a time-spacedescription in deformed Lorentz metrics and three dimensional fractal structures [9,10].

This algebraic structure has two representations, an additive and a multiplicativerepresentation that correspond to the rectangular and polar versions of complex numbers.The relationship between the two scator representations is established by a generalizedEuler’s type equation. The components exponential function is a scator function of scatorargument that extends the notion of the complex exponential to higher dimensions [11].This function, labeled ’cexp’, is scator holomorphic and its derivative is the function

itself, thus it satisfies the differential equation f ′(oζ)= f

(oζ), where the prime denotes

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 31 August 2021 doi:10.20944/preprints202108.0572.v1

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differentiation with respect tooζ. The additive and multiplicative (rectangular/polar) scator

representations allow for a generalization of De Moivre’s formula to 1 + n dimensions inscator algebra.

In this paper, the essentials of elliptic scator algebra are presented in section 2. Em-phasis is laid on the transformations between representations. Just as in real and complexalgebra, the components exponential function, reviewed in this section, maps scator ad-dition onto scator multiplication. This result is used to prove the commutative groupproperties in the multiplicative representation and its relationship with non-associativity inthe additive representation. The relation between a scator raised to an integer power andangle multiplication is established in section 3. A geometric representation of these resultsis presented in section 4. The exponential of a 1 + 2 dimensional scator and its relationshipwith the components exponential function, is obtained in section 5.

2. Elliptic scator algebra

In the additive representation, scator elements are represented by a sum of components,

oϕ = f0 +

n

∑j=1

f jej, (1)

where f0, f j ∈ R, j from 1 to n ∈ N and ej /∈ R. The scator coefficients in this representation,named the additive variables, are tagged with lowercase Latin letters. The multiplicative orpolar representation of a scator is

oϕ = ϕ0

n

∏j=1

eϕj ej = ϕ0

n

∏j=1

exp(

ϕjej). (2)

where ϕ0 ∈ R+, ϕj ∈ R, and ej /∈ R; R+ represents the interval [0, ∞) . The scator variablesin the multiplicative representation ϕ0, ϕj, named multiplicative variables, are with Greekletters. The zero subindex component is the scalar component in either representation. Thedirector components are labeled with subindices 1 to n. For all elements in the S1+n scatorset, the additive scalar component must be different from zero if two or more directorcomponents are not zero,

S1+n ={ o

ϕ = f0 +n

∑j=1

f jej : f0 6= 0 if ∃ f j fl 6= 0, for j 6= l from 1 to n}

. (3)

In the early papers E1+n was used instead of S1+n.

2.1. Transformation between representations

The transformation from the multiplicative to additive representation originates fromEuler type relationships for each ej director component, eϕj ej = cos ϕj + sin ϕj ej, where

the scator product of a unit director with itself is minus one, ejej = −1. The scatoroϕ given

by (2), is thenoϕ = ϕ0

n

∏j=1

eϕj ej 7−→ ϕ0

n

∏j=1

(cos ϕj + sin ϕj ej

). (4a)

The product of terms involving only orthogonal director components in the factors isrequested to distribute over addition. In general, scator multiplication does not distributeover addition [7]. Two unit scators are orthogonal in scator algebra, if their scator productis zero ejek = 0. The n director units are requested to be orthogonal between them, that is,ejek = 0 for all j 6= k, for j, k from 1 to n. The product of the scator components is then

oϕ = ϕ0

n

∏j=1


)= ϕ0

n

∏k=1

cos(ϕk) + ϕ0

n

∑j=1

n

∏k 6=j

cos(ϕk) sin(

ϕj)ej. (4b)



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The mapping of the multiplicative to additive representations is given by the functionfmar : (R+;Rn)→ S1+n,

fmar : ϕ0

n

∏j=1

eϕj ej 7−→ ϕ0

n

∏k=1

cos(ϕk)︸︷︷︸f0

+ ϕ0

n

∑j=1

n

∏k 6=j

cos(ϕk) sin(

ϕj)

︸︷︷︸f j

ej. (5)

The arguments of the trigonometric functions are of course, modulus 2π due to theirfundamental period. The inverse mapping is given by the function famur : S1+n → (R+;Rn)defined, for scators with f0 6= 0, by

famur : f0 +n

∑j=1

f jej 7−→ | f0|n

∏j=1

√√√√1 +f 2j

f 20︸︷︷︸

ϕ0

exp(

arctan( f j

f0

)︸︷︷︸

ϕj

ej

), (6a)

for scators with zero additive scalar component and non-zero ej director component by

famur : f jej 7−→∣∣ f j∣∣ exp

(sgn(

f j)π

2ej

), (6b)

and for the null scator

famur : 0 +n

∑j=1

0 ej 7−→ 0n

∏j=1

exp(0 ej). (6c)

To recap, there are two representations of scator numbers, additive and multiplicative,each of them with their corresponding additive (1) or multiplicative (2) variables. A scatornumber can also be expressed in the additive representation with multiplicative variables(5) or in the multiplicative representation with additive variables (6a),(6b). These fourpossibilities are a higher dimensional analogue of complex numbers that can be writtenin Cartesian form with real and imaginary parts z = a + ib, or polar form z = reiϕ withmodulus and angle variables; z = r(cos ϕ + i sin ϕ) or z =

√a2 + b2 exp

(i arctan

( ba))

being the other two possible ways to write them. The relationships between variables are

f0 = ϕ0

n

∏k=1

cos ϕk; f j = ϕ0

n

∏k 6=j

cos ϕk sin ϕj. (7a)

And

ϕ0 = | f0|n

∏j=1

√√√√1 +f 2j

f 20

, if f0 6= 0, ϕ0 =∣∣ f j∣∣ if f0 = 0, ϕ0 = 0 if f0 = f j = 0 ∀j; (7b)

ϕj = arctan( f j

f0

). (7c)

There is no scator additive to multiplicative representation in the R1+n \S1+n set [7, Lemma4.1]. In 1 + 1 dimensions, there are j subsets S1+1

j ⊆ S1+n where all el coefficients vanish,

except for the jth component. These j scator sets are isomorphic to C [12, Proposition4.12]. Equation (5) reduces to Euler’s formula for any one of these sets, fmar : ϕ0eϕl el 7−→ϕ0 cos(ϕl) + ϕ0 sin(ϕl)el , where el /∈ R represents a hyper-imaginary unit, el el = −1.



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2.2. Scator algebra - fundamental operations

The sum for scatorsoα = a0 + ∑n

j=1 ajej ∈ R1+n andoβ = b0 + ∑n

j=1 bjej ∈ R1+n is

oα +

oβ = (a0 + b0) +

n

∑j=1

(aj + bj

)ej. (8)

Scator addition satisfies Abelian group properties.

2.3. Product in the additive representation

S1+n is the subset of R1+n where multiplication is defined in the additive representa-tion.

Definition 1. Foroα = a0 + ∑n

j=1 ajej ∈ S1+n andoβ = b0 + ∑n

j=1 bjej ∈ S1+n, the productoα

oβ

is: if a0b0 6= 0,

oα

oβ = a0b0

n

∏k=1

(1− akbk

a0b0

)+ a0b0

n

∑j=1

[n

∏k 6=j

(1− akbk

a0b0

)( aj

a0+

bj

b0

)]ej; (9a)

If a0 = 0 and b0 6= 0,oα = al el has a single nonvanishing director component,

oα

oβ = −albl + alb0el −

n

∑j 6=l

(al

blbj

b0

)ej; (9b)

If a0 = b0 = 0,oα = al el and

oβ = bjej,

oα

oβ = −albjδl j, (9c)

where δl j is a Kroneker delta. The (9b), (9c) product definitions can be obtained from the appropriatelimits of (9a). Evaluation of the director components limits should always be performed priorto the scalar component limit, in order to remain within the S1+n set [8]. Conjugation of

oϕ =

f0 + ∑nj=1 f jej is defined by

oϕ∗≡ f0 −∑n

j=1 f jej. The magnitude is given by the positive square

root of the scator times its conjugate∥∥ o

ϕ∥∥ =

√oϕ

oϕ∗. For a scator with non vanishing additive

scalar component ∥∥ oϕ∥∥ =

√oϕ

oϕ∗=∣∣ f0∣∣ n

∏k=1

√1 +

f 2k

f 20

, (10)

and if the additive scalar component is zero there is only one director component, say the lth

component,∥∥ o

ϕ∥∥ =

∣∣ fl∣∣. The multiplicative inverse of a scator

oϕ is

oϕ−1

=oϕ∗/∥∥ o

ϕ∥∥2.

Multiplication is commutative and, if zero is avoided, all elements have inverse inS1+n. A previous communication has been devoted to the conditions when the product isor is not associative in the additive representation [8]. The two main results are:

Theorem 1 (Fernandez-Guasti, 2018). “The scator product in S1+n is associative in the additive

representation if all possible product pairs have a non vanishing additive scalar component,( oα

oβ) o

ϕ =oα( o

βoϕ)=

oβ( oα

oϕ)

ifoα

oβ,

oβ

oϕ,

oα

oϕ ∈ S1+n

6=0 .”

The set S1+n6=0 ⊆ S1+n is the subset of S1+n with non zero additive scalar component.



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Theorem 2 (Fernandez-Guasti, 2018). The scator product in S1+n is in general not associativefor n > 1 in the additive representation, if one or more of the possible product pairs has a vanishing

additive scalar component,( oα

oβ) o

ϕ 6= oα( o

βoϕ)6=

oβ( oα

oϕ)

ifoα

oβ or

oβ

oϕ or

oα

oϕ ∈ S1+n

0 .

Elements in S1+n0 can only have a single non vanishing director component, say the

j component from the n director components, f jej ∈ S1+nj . The set of elements that have

zero additive scalar component is S1+n0 =

⋃nj=1 S

1+nj . In the associative case, for n > 1, all

products invoke (9a), whereas in the non-associative case, a product rule other than (9a) isinvoked at least in one of the products. Theorem (1) establishes a sufficient condition forthe scator product to be associative in the additive representation, whereas Theorem (2)establishes a necessary condition for non-associativity.

2.4. Product in the multiplicative representation

In the multiplicative representation, the product of two scators is evaluated by per-forming the multiplicative scalars product and the addition of the multiplicative directorcoefficients with the same director unit,

oα

oβ =

(α0

n

∏j=1

exp(αjej

))(β0

n

∏j=1

exp(

β jej))

= α0β0

n

∏j=1

exp[(

αj + β j)ej]. (11)

The multiplicative scator components having the same director ej, satisfy the addition

theorem for exponents. The conjugate of the scatoroϕ = ϕ0 ∏n

j=1 eϕj ej , is obtained by the neg-ative of the director components, leaving the multiplicative scalar component unchangedoϕ∗≡ ϕ0 ∏n

j=1 e−ϕj ej . The magnitude of a scator∥∥ o

ϕ∥∥ =

√oϕ

oϕ∗, in the multiplicative repre-

sentation from (11), is∥∥ o

ϕ∥∥ = ϕ0. The multiplicative scalar represents the scator magnitude.

The multiplicative inverseoϕ−1

=oϕ∗∥∥ o

ϕ∥∥−2 exists if the scator magnitude is not zero.

The product in the multiplicative representation is closely related to the componentsexponential function [11]. The

ocexp function is a scator function

ocexp : R1+n → S1+n of

scator variable, sayoζ = z0 + ∑n

j=1 zjej, given in the multiplicative representation, by theproduct of the exponential of each component,

ocexp

(oζ)≡ o

cexp(

z0 +n

∑j=1

zjej

)≡ exp(z0)

n

∏j=1

exp(zjej

). (12a)

Theocexp function is scator holomorphic in the entire R1+n domain according to the differ-

ential quotient criterion [11]. In the additive representation, theocexp function is

ocexp

(z0 +

n

∑j=1

zjej

)≡ ez0

n

∏k=1

cos(zk)︸︷︷︸cexp0

(oζ)

+n

∑j=1

ez0n

∏k 6=j

cos(zk) sin(zj)

︸︷︷︸cexpj

(oζ)

ej. (12b)

Lemma 1. Theocexp : R1+n → S1+n function maps scator addition onto scator multiplication.

Proof. For scatorsoα +

oβ,

ocexp

( oα +

oβ)= exp(a0 + b0)

n

∏j=1

exp((

aj + bj)ej).



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Since the exponential addition theorem is satisfied component-wise,

ocexp

( oα +

oβ)= exp(a0) exp(b0)

n

∏j=1

exp(ajej

)exp

(bjej

),

this expression can be rearranged to obtain

ocexp

( oα +

oβ)=

ocexp

( oα) o

cexp( o

β). (13)

The multiplicative scator representation in terms of theocexp function is

oϕ = ϕ0

ocexp

( n

∑j=1

ϕjej

), (14)

as can be seen from (2) and (12a).

Corollary 1. The scator product satisfies commutative group properties in the multiplicativerepresentation.

Proof. The commutative group properties of addition for the director components aremapped onto the commutative group properties of the director components in the multi-plicative representation. The product of the multiplicative scalar components is a product oftwo real numbers; provided that the scator with zero magnitude is excluded, commutativegroup properties are satisfied.

In particular, product associativity is satisfied in the multiplicative representation,

oα

oβ

oϕ =

( oα

oβ) o

ϕ =oα( o

βoϕ)=

oβ( o

ϕoα)

=

(α0

n

∏j=1

exp(αjej

))(β0

n

∏j=1

exp(

β jej))(

β0

n

∏j=1

exp(

β jej))

= α0β0 ϕ0

n

∏j=1

exp[(

αj + β j + ϕj)ej].

Remark 1. The product of non zero factors is never zero in the multiplicative representation, forexample, let αj + β j =

π2 for all j from 1 to n,

oα

oβ =

(α0

n

∏j=1

exp(αjej

))(β0

n

∏j=1

exp(

β jej))

= α0β0

n

∏j=1

exp(π

2ej

).

The magnitude of this scator is∥∥ o

αoβ∥∥ =

(α0β0 ∏n

j=1 exp(

π2 ej)α0β0 ∏n

j=1 exp(−π

2 ej)) 1

2= α0β0.

In the multiplicative representation, the product of the magnitudes is always equal to the magnitudeof the products.

Definition 2. The product of exponentials with different ej components is irreducible in themultiplicative representation, that is, it is not possible to express such a scator with fewer factors.The product φ0 ∏n

j=1 exp(

ϕjej)

can no longer be simplified, regardless of the values of the ϕj

coefficients. Inasmuch as the sum f0 + ∑nj=1 f jej, is irreducible in the additive representation, for it

cannot be written with fewer addends.



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2.5. Product in the additive representation with multiplicative variables

The product can be written with multiplicative variables in the additive representationfrom (5) and (11),

oα

oβ = α0β0

n

∏j=1

e(αj+β j)ej 7−→

α0β0

n

∏k=1

cos(αk + βk) + α0β0

n

∑j=1

n

∏k 6=j

cos(αk + βk) sin(αj + β j

)ej. (15)

The conditions for a product to give a zero scalar component or a null scator in terms ofthe additive variables have been discussed before [8, Sec.3.1]. Consider these conditionsusing multiplicative variables.

Zero additive scalar component: If any one of the cosine arguments is zero, the scalar

component ofoα

oβ is zero. Let it be the argument of the lth component

cos(αl + βl) = cos(π

2

)= 0,

This is condition a0b0 = albl in additive variables, since 1 = albla0b0

= tan αl tan βl , andcos αl cos βl = sin αl sin βl can be rewritten as the cosine of the sum of angles, cos(αl + βl) =0. A particular value of the previous case is αl = π

2 , then βl = 0, that corresponds, in

additive variables, tooα = al el when the el component of

oβ is zero, bl = 0.

Zero products: If two or more director arguments are equal to π2 mod π, a scator is

zero in the additive representation. Let the el and eq arguments be π2 , from (15), the scalar

component and all director components with j 6= l, q involve a cos2(π2)

factor, the el andeq coefficients involve a cos

(π2)

factor,

oα

oβ = α0β0

n

∏k 6=l,q

cos(αk + βk) cos2(π

2

)+ α0β0

n

∑j=1

n

∏k 6=j,l,q

cos(αk + βk) cos2−δjl−δjq(π

2

)sin(αj + β j

)ej = 0 +

n

∑j=1

0 ej. (16)

The multiplicative (11) and additive (9a)-(9c) definitions of the product are equivalent.In the early papers, this equivalence led us to a wrong statement regarding the productassociativity in the additive representation, that has now been corrected [8]. The lack ofassociativity in the additive representation arises because the single multiplication rule inthe polar representation is mapped into three different rules in the additive representation.When different rules in the additive representation have to be used, product associativitybecomes an issue. Some examples will be given in the next sections.

3. Victoria equation

The exponential addition theorem is satisfied component wise in the multiplicativerepresentation of scator algebra, thus the following proposition.

Theorem 3. In the multiplicative representation, for a scatoroϕ ∈ S1+n raised to the power m ∈ Z,

the exponent m distributes over the scator factors

oϕ

m=

(ϕ0

n

∏j=1

eϕj ej

)m

= ϕm0

n

∏j=1

emϕj ej . (17)



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Proof. From (11), ifoα =

oβ

oα

2=

(α0

n

∏j=1

eαj ej

)(α0

n

∏j=1

eαj ej

)= α2

0

n

∏j=1

e2αj ej ,

by induction, for m ∈ N, the mth power of a scator in the multiplicative representation is

oϕ

m=

(ϕ0

n

∏j=1

eϕj ej

)m

= ϕm0

n

∏j=1

emϕj ej . (18)

The multiplicative inverse is

oϕ−1

=∥∥ o

ϕ∥∥−2 o

ϕ∗= ϕ−1

0

n

∏j=1

e−ϕj ej .

Replace ϕ0 → ϕ−10 and ϕj → −ϕj in (18),(

oϕ−1)m

=

(ϕ−1

0

n

∏j=1

eϕj ej

)m

=(

ϕ−10

)m n

∏j=1

em(−ϕj)ej ,

but since ϕ0, ϕj ∈ R,(

ϕ−10

)m= ϕ−m

0 and m(−ϕj

)= −mϕj, thus

oϕ−m

= ϕ−m0 ∏n

j=1 e−mϕj ej .

Therefore (18) is satisfied for integer m, where( o

ϕ)0 ≡

01.

Since the scator product is associative in the multiplicative representation, there is no

need to place an explicit association in the products, i.e.oϕ

3=

oϕ

oϕ

oϕ. For example, consider

the unit magnitude scatoroϕ = exp

(π4 e1)

exp(2 e2), thenoϕ

3= exp

( 3π4 e1

)exp(6 e2).

The multiplicative to additive transformation together with Theorem 3, allow for thefollowing generalization of De Moivre formula in scator algebra.

Theorem 4. Provided that the product of the factors and its components are associative, a scatoroϕ ∈ S1+n raised to the power m ∈ Z, can be written in the additive representation as

(ϕ0

n

∏k=1

cos ϕk +n

∑j=1

ϕ0

n

∏k 6=j

cos ϕk sin ϕj ej

)m

= ϕm0

n

∏k=1

cos(mϕj

)+

n

∑j=1

ϕm0

n

∏k 6=j

cos(mϕk) sin(mϕj

)ej. (19)

Proof. From the multiplicative to additive transformation (5) applied to both sides ofEq. (17), Eq. (19) is obtained. While (17) is valid for any arguments, the possible lackof associativity in the additive representation could mar (19), because the left side ofthe equation is product associativity dependent while the right side is not. Theorem 1establishes the associativity sufficient condition in the additive representation, namely, thatnone of the possible product associations give a scator with zero scalar additive componentfor n > 1. The multiplicative to additive transformation (5), involves the product of thescator components (4a). Equation (19) in terms of the director components factors is(

ϕ0

n

∏j=1


))m

= ϕm0

n

∏j=1

(cos(mϕj

)+ sin

(mϕj

)ej

). (20)

This equation and thus Eq. (19) is satisfied if none of the m× n products on the left of (20),produce a scator with zero additive scalar component for n > 1. If there is only a single



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one non-zero director component, the scator product becomes a bilinear operation in theadditive representation (9a)-(9c), associativity of the product is then insured even if theadditive scalar component is zero.

For historical reasons, we shall refer to (19) as the victoria equation.

Example 1. Consider the victoria equation in 1 + 2 dimensions with m = 2,

(cos ϕ1 cos ϕ2 + cos ϕ2 sin ϕ1 e1 + cos ϕ1 sin ϕ2 e2)2 =

cos(2ϕ1) cos(2ϕ2) + cos(2ϕ2) sin(2ϕ1)e1 + cos(2ϕ1) sin(2ϕ2)e2, (21)

where ϕ20 has been canceled on both sides of the equation. The squaring operation on the LHS of

(21) can be evaluated from (9a),

( f0 + f1e1 + f2e2)2 = f 2

0

(1−

f 21

f 20

)(1−

f 22

f 20

)+ 2 f0 f1

(1−

f 22

f 20

)e1

+ 2 f0 f2

(1−

f 21

f 20

)e2, (22)

replacement in terms of the multiplicative variables from comparison with (21) gives

(cos ϕ1 cos ϕ2 + cos ϕ2 sin ϕ1 e1 + cos ϕ1 sin ϕ2 e2)2

= cos2 ϕ1 cos2 ϕ2

(1− tan2 ϕ1

)(1− tan2 ϕ2

)+ 2 cos ϕ1 cos2 ϕ2 sin ϕ1

(1− tan2 ϕ2

)e1 + 2 cos ϕ2 cos2 ϕ1 sin ϕ2

(1− tan2 ϕ1

)e2.

This result is substituted in (21), and equating term to term:scalar components:

(cos2 ϕ1 − sin2 ϕ1

)(cos2 ϕ2 − sin2 ϕ2

)= cos(2ϕ1) cos(2ϕ2),

e1 components: 2 cos ϕ1 sin ϕ1(cos2 ϕ2 − sin2 ϕ2

)= cos(2ϕ2) sin(2ϕ1),

and the e2 equality: 2 cos ϕ2 sin ϕ2(cos2 ϕ1 − sin2 ϕ1

)= cos(2ϕ1) sin(2ϕ2).

These expressions are products of familiar double angle trigonometric identities.

Definition 3. If the victoria equation (19), is evaluated symbolically, that is, without enteringany specific values for the arguments, the (9a) product rule should always be used. If thereafter,particular scator component values are inserted, the procedure is known as the postevaluationalgorithm.

The victoria equation is satisfied if the postevaluation algorithm is adopted, evenif associativity of the factors is not fulfilled. In this case, associativity in the additiverepresentation should not have any precedence parenthesis if the product is evaluatedsymbolically prior to entering specific values.

Example 2. Evaluateoϕ

3from repeated use of (9a),

oϕ

3=

oϕ

oϕ

oϕ = ( f0 + f1e1 + f2e2)

3 = f 30

(1−

3 f 21

f 20

)(1−

3 f 22

f 20

)

+ f1

(1−

3 f 22

f 20

)(3 f 2

0 − f 21

)e1 + f2

(1−

3 f 21

f 20

)(3 f 2

0 − f 22

)e2. (23)

For ϕ0 = 1, ϕ1 = π4 , ϕ2 = 2, the additive variables are f0 = f1 = 1√

2cos(2), and f2 = 1√

2sin(2).

The postevaluated expression, from (23) is ( f0 + f1e1 + f2e2)3 ≈ −0.679 + 0.679 e1 + 0.198 e2.



10 of 22

The same result is obtained from the RHS of (19),− 1√2

cos(6) + 1√2

cos(6)e1− 1√2

sin(6)e2. Theoutcome is identical in the multiplicative representation. The postevaluation procedure avoids in

this case, the zero scalar singularity. However, evaluateoϕ

2=

oϕ

oϕ in the additive representation

with the specific values. Since f0 = f1 from (9a),oϕ

2=( o

ϕoϕ)= 2

(f 20 − f 2

2)e1. The product( o

ϕoϕ) o

ϕ, since( o

ϕoϕ)

has zero additive scalar component, has to be evaluated with (9b),

( oϕ

oϕ) o

ϕ =(

2(

f 20 − f 2

2

)e1

)( f0 + f0e1 + f2e2)

= 2(

f 20 − f 2

2

)(− f0 + f0e1 − f2e2),

an altogether different expression is obtained compared withoϕ

3=

oϕ

oϕ

oϕ given by (23). The

numerical value of the non associative product in this example is( o

ϕoϕ) o

ϕ ≈ −0.19 + 0.19 e1 −0.42 e2. If the products are not associative, the way the m products in

oϕ

mare associated, has to be

stated explicitly.

Corollary 2. If the coefficients of the director components in multiplicative variables of a scatoroϕ

are algebraic real numbers, thenoϕ

m, m ∈ Z is associative in the additive representation.

Proof. The multiplicative coefficients ϕj of a scator raised to the power m ∈ Z are mϕj.Since the product mϕj is also algebraic but π is transcendental, mϕj cannot be equal to π

2 .At least one component mϕj =

π2 is necessary for a scator product to have a null scalar

component. This in turn, is a necessary condition for non-associativity.

Corollary 3. If none of the director coefficients in multiplicative variables of a scatoroϕ is of the

form π2p , |p| > |m|, p, m ∈ Z, then

oϕ

m, is associative in the additive representation.

Proof. This result follows from the victoria equation (19), the zero additive scalar compo-nent condition in subsection 2.5 and Thm. 1.

3.1. Nilpotent and nil-scalar-potent elements

Elements whose three components have equal absolute value in the additive repre-sentation, are the only non trivial square nilpotent elements in 1+2 dimensional imaginaryscator algebra [10, Lemma 1]. The present formalism permits a more encompassing propo-sition.

Lemma 2. A scatoroϕ = ϕ0 ∏n

k=1 cos(ϕk) +∑nj=1 ϕ0 ∏n

k 6=j cos(ϕk) sin(

ϕj)ej, is power m nilpo-

tent in the additive representation, if ϕl = ϕq = ± π2m mod π, for any two different director

components, l 6= q and none of the ϕj’s different from ϕl , ϕq, are of the form π2p with p < m.

Proof. The scator

oϕ = ϕ0

n

∏k 6=l,q

cos(ϕk) cos2( π

2m

)+ ϕ0

n

∑j=1

n

∏k 6=j

cos(ϕk) cos( π

2m

)sin(

ϕj)ej,

evaluated to the mth power from (19) is,

oϕ

m= ϕm

0

n

∏k 6=l,q

cos(mϕk) cos2(π

2

)+ ϕm

0

n

∑j 6=l

n

∏k 6=j,l,q

cos(mϕk) cos2−δjl−δjq(π

2

)sin(mϕj

)ej.



11 of 22

But all terms involve at least a cos(

π2)= 0 factor, thus

oϕ

m= 0. The condition on the ϕj 6=l,q

director coefficients insures that the scator is not nilpotent for any smaller m.

Definition 4. A scatoroϕ is power m nil-scalar potent if the additive scalar component of

oϕ

mis

zero.

Corollary 4. A scatoroϕ ∈ S1+n \ S1+1

j is power m nil-scalar potent, if ϕl = ± π2m mod π and

none of the ϕj’s different from ϕl are of the form π2p with p < m. The products of

oϕ

m+1are not

associative in the additive representation.

Proof. The proof is analogous to Lemma (2), but only one director component is requestedto satisfy ϕl = ± π

2m mod π,

oϕ = ϕ0

n

∏k 6=l

cos(ϕk) cos( π

2m

)+ ϕ0

n

∑j=1

n

∏k 6=j

cos(ϕk) sin(

ϕj)ej,

then, the el director component will not vanish

oϕ

m= ϕm

0

n

∏k 6=l

cos(mϕk) sin(±π

2

)el .

Sinceoϕ has two or more non zero director components and

oϕ

mhas vanishing additive

scalar component, from Thm. 2, the productoϕ

m oϕ is in general, not associative.

It is not straight forward to state whether a scator number in the additive representa-

tion with additive variables, i.e.oϕ = f0 + ∑n

j=1 f jej, is nil-scalar-potent for some exponentm. A possible route to give an answer without a direct calculation is to evaluate the mul-

tiplicative director coefficients via (7a), ϕj = arctan( f j

f0

). An m nil-scalar-potent scator

requires that ϕj = ± π2m for any j from 1 to n. Thus, for each component, evaluate

π

2 arctan( f j

f0

) = m, if m ∈ Z, (24)

if m is an integer, the scator will be power m nil-scalar-potent. It is of course possible thatdifferent director components have different m’s that satisfy (24), then for each mj ∈ Z thescator will be power mj nil-scalar-potent. If two or more components satisfy the condition(24), the scator is power m nilpotent.

Example 3. Consider the scator,

oϕ = f0 + f0

n

∑j 6=l,q

tan(

ϕj)ej + f0

(2−√

3)

el − f0

(2−√

3)

eq. (25)

where f0 = ∏nk=1 cos(ϕk). In the additive representation, identify the different director components

with equal (absolute value) coefficients. For each pair evaluate

π

2 arctan(

flf0

) =π

2 arctan(

2−√

3) =

π

2 π12

= 6.

Thus the scator (25) is power 6 nilpotent.



12 of 22

In contrast with split-quaternions, only the identity element is idempotent in elliptic

scators. To prove it, consideroϕ

2=

oϕ in the multiplicative form

ϕ20

n

∏j=1

e2ϕj ej = ϕ0

n

∏j=1

eϕj ej ,

ϕ20 = ϕ0 only for ϕ0 = 1, and ϕj = 2ϕj only if ϕj = 0, thus only the identity is idempotent.

4. Geometric interpretation

The scalar component and two director components can be depicted in orthogonaldirections in a three dimensional plot. The director coefficients in multiplicative variablesϕx and ϕy are geometrically represented by the angle between the scalar (real) axis andthe projection onto the corresponding hyperimaginary director axis as shown in figure1. Notice that these angles are neither spherical coordinate angles nor direction cosines,nor Euler angles. Positive scator angles are measured taking the scalar axis as referenceand measuring towards the corresponding positive director axis. In low dimensional cases,letters are sometimes used for the director coefficient subindices instead of numbers.

Figure 1. S1+2 scator geometrical representation in three dimensional space. The scalar and twodirector components are depicted in orthogonal axes. The red line shows the intersection of theoϕ(1; ϕx, 0), ey plane (yellow) and the

oϕ(1; 0, ϕy

), ex plane (green). The constant scator magnitude

curve is represented by the continuous gray segment.

The product of two scators with constant coefficients involves a scaling and a rotationin each scalar-hypercomplex plane. Consider the product of unit magnitude scators sothat scaling is not an issue in the following discussion. The magnitude of a product isequal to the product of the magnitudes provided that the products are associative [8].Therefore, for n > 1, the product of unit magnitude scators will remain unitary if thescalar component of the products and its scator conjugates do not vanish. In particular, thepowers of associative unit scators lie on the isometric scator surface, called a cusphere [11].The unit cusphere is an n dimensional surface embedded in a 1 + n dimensional space that

satisfies the condition∥∥ o

ϕ∥∥ =

o1. The additive representation with multiplicative variables

of a unit scator isoϕ =

n

∏k=1

cos ϕk +n

∑j=1

n

∏k 6=j

cos ϕk sin ϕjej,

where ϕ0 = 1. Ifoϕ ∈ S1+2,

oϕ(1; ϕx, ϕy

)= cos ϕx cos ϕy + cos ϕy sin ϕx ex + cos ϕx sin ϕy ey, (26)



13 of 22

where, on the left, the scator components are explicitly shown, the scalar component isseparated by a semicolon from the director components that are separated by commas.Recall that Greek letters with subindices represent multiplicative variables whereas Latin

letters, i.e.oϕ(

f0; fx, fy), represent additive variables. The scator (26) can be factored as

oϕ = (cos ϕx + sin ϕx ex) cos ϕy + cos ϕx sin ϕyey, so that the angle ϕx in the s, ex projection(the scalar axis is labeled s for short) is preserved regardless of the ϕy angle. The plane

defined by the unit scatorsoϕ(1; ϕx, 0) and ey is depicted in yellow in figure 1. Similarly,

oϕ = cos ϕy sin ϕx ex +

(cos ϕy + sin ϕyey

)cos ϕx, evinces that the angle in s, ey is preserved

regardless of the ϕx angle, as shown in figure 1. The plane defined by the scatorsoϕ(1; 0, ϕy

)and ex is depicted in green. The scator (26) can then be viewed as the intersection of thesetwo planes.

Lemma 3. There is a unique decomposition of a scator in terms of its magnitude and the product ofunit scators with single non-vanishing orthogonal director components.

Proof. The proof follows from Eq. (4b) proceeding in the inverse direction. The multi-plicative scalar ϕ0 represents the scator magnitude. If ϕ0 is factored, the remaining unitmagnitude scator in S1+n is

n

∏k=1

cos(ϕk) +n

∑j=1

n

∏k 6=j

cos(ϕk) sin(

ϕj)ej.

Each ej director component can be factored with an Euler type expression(cos ϕj + sin ϕj ej

),

oϕ = ϕ0

n

∏j=1


).

Since ϕj = arctan( f j

f0

)and ϕ0 = | f0|

n∏j=1

√1 +

f 2j

f 20

, an arbitrary scator in additive variables

oϕ = f0 + ∑n

j=1 f jej ∈ S1+n, can then be factored as

oϕ = | f0|

n

∏j=1

√√√√1 +f 2j

f 20

n

∏j=1

(cos(

arctan( f j

f0

))+ sin

(arctan

( f j

f0

))ej

).

The products in the above expression can be evaluated to recover the original scator,

cos(

arctan( f j

f0

))=

Sgn(cos ϕj

)| f0|√

f 20 + f 2

j

=Sgn

(cos ϕj

)√1 +

f 2j

f 20

and

sin(

arctan( f j

f0

))=

f j√f 20 + f 2

j

=f j/| f0|√1 +

f 2j

f 20

,

thusoϕ = | f0|

n

∏j=1

(Sgn

(cos ϕj

)+

f j

| f0|ej

).



14 of 22

This product can then be readily written as a sum for scators since all director componentsare orthogonal [11, Eq.(12)],

oϕ = | f0|

(Sgn( f0) +

n

∑j=1

f j

| f0|ej

),

where Sgn( f0) = ∏nj=1 Sgn

(cos ϕj

). Thus

oϕ = | f0|Sgn( f0) +

| f0|| f0|

n

∑j=1

f j ej = f0 +n

∑j=1

f jej.

The S1+n scator decomposition can be geometrically represented as the projection onto nplanes. Each of these

(s, ej

)∈ S1+1

j planes, share the scalar axis but have a single directorcomponent orthogonal to the remaining director units.

The projections of the S1+2 dimensional scator (26), depicted in figure 1 are,

oϕ(1; ϕx, 0) = cos ϕx + sin ϕx ex,

oϕ(1; 0, ϕy

)= cos ϕy + sin ϕyey.

Scator elements (unlike other objects like vectors) can be written as the product of its

projections; In this example,oϕ(1; ϕx, ϕy

)=

oϕ(1; ϕx, 0)

oϕ(1; 0, ϕy

). Notice that the 1 + 2 D

scator as well as its 1 + 1 D projections are unitary scators. This is only possible becausethe scator magnitude (10) is not a quadratic Euclidean form.

Figure 2. Geometrical representation of multiplication of the unit scatorsoα (red) times

oβ (blue). Their

product is theoϕ =

oα

oβ scator (green), where the scator angles are ϕx = αx + βx and ϕy = αy + βy. All

scators as well as their projections onto the planes have unit magnitude. (qualitative drawing)

The representation (26) lends nicely to parametrically plot the cusphere as shown

in figure 3. The square magnitude ofoϕ(1; ϕx, ϕy

)in terms of additive components, from

(10), is∥∥ o

ϕ∥∥2

=o1

2= cos2 ϕx cos2 ϕy

(1 + tan2 ϕx

)(1 + tan2 ϕy

). The Pythagorean identity

in S1+2 scator space is

1 =(

sin2 ϕx + cos2 ϕx

)(sin2 ϕy + cos2 ϕy

).

The geometric interpretation of the product for arbitrary unit scators is rather rich andcomplex. The product requires the sum of scator angles in each direction measured with



15 of 22

respect to the scalar axis. A qualitative geometrical representation is depicted in figure 2.Notice that both angles are involved in each additive component as evinced in (11).

In S1+2, there are three degrees of freedom, f0; fx, fy in additive variables or ϕ0; ϕ1, ϕ2

in multiplicative variables. Unit magnitude scators∥∥ o

ϕ∥∥ =

o1 = ϕ0, leave two independent

quantities: In multiplicative variables, the director coefficients ϕx, ϕy, whereas the additive

variables are related by 1 = f−20(

f 20 + f 2

x)(

f 20 + f 2

y

). The

oϕ ∈ S1+2 unit scator raised to the

power m, from the victoria equation is

oϕ

m= cos(mϕx) cos

(mϕy

)+ cos

(mϕy

)sin(mϕx)ex + cos(mϕx) sin

(mϕy

)ey. (27)

The product of unit scators in the scalar s, ex plane or the s, ey plane, represents properrotations in these planes. This is expected since S1+1

j is isomorphic to complex algebra forany one j from 1 to n. The novel part comes from scators that do not lie on these planes.

4.1. Scators with equal hypercomplex components

Consider to begin with, unit magnitude scators with equal hypercomplex directorcomponents, ϕx = ϕy, from (26)

oϕ = cos2 ϕx +

12

sin(2ϕx)ex +12

sin(2ϕx)ey.

This scator lies on the 45 degree plane with respect to the ex, ey directions. The square ofthis scator is

oϕ

2= cos2(2ϕx) +

12

sin(4ϕx)ex +12

sin(4ϕx)ey,

that again lies on the π4 director-director plane and must be on the cusphere surface, since

unit magnitude invariance holds if associativity is satisfied. The same is true for higherorder powers,

oϕ

m= cos2(mϕx) +

12

sin(2mϕx)ex +12

sin(2mϕx)ey.

Thus, all integer powers of a scator with equal director components lie on the equaldirector’s coefficients plane. For example, let ϕx = ϕy = π

6 ,

oϕ = cos2

( π

12

)+

12

sin(π

6

)ex +

12

sin(π

6

)ey,

that in additive variables is

oϕ =

14

(2 +√

3)+

14

ex +14

ey.

The angles in the ex and ey directions increase by π6 every time a product with

oϕ is

performed, the 12 + 1 power returns to the original positionoϕ

13=

oϕ. The integer powers

of this scator are depicted in figure 3. For a scator where the ϕx director coefficient isa trigonometric number, i.e. equal to a rational multiple of π, the powers will repeat.

Since all trigonometric numbers are algebraic numbers [13], the power of scatorsoϕ =

f0 + fx ex + fx ey with real algebraic additive coefficients f0, fx will eventually repeat. Theconverse is also true, if a director coefficient in the multiplicative representation is not a

rational multiple of π,oϕ

mwill never be equal to

oϕ for integer m. Consider the curve where

all powers of unit scators with equal directors lie, be them eventually periodic or not. Tothis end, let t = ϕx = ϕy, the parametric curve is

oϕ(1; t, t) =

12+

12

cos(2t) +12

sin(2t)ex +12

sin(2t)ey. (28)



16 of 22

Figure 3. Powers of scators with equal director coefficients lie on the continuous curve (red), givenby (28). This curve lies on the plane at 45 degrees with respect to the ex, ey axes. Powers of the

scatoroϕ = cos2( π

12)+ 1

2 sin(

π6)ex +

12 sin

(π6)ey are shown in blue dots. The cusphere surface is the

constant magnitude scator surface depicted in a semitransparent shade.

This function is mod π periodic since all arguments involve the double angle. The projec-tions in the s, e1 plane and the s, e2 plane are circles with radius 1

2 centerd at s = 12 . The

curve is centered at s = 12 and at this s, the maximum value of the director components is

obtained. The curve is shown in figure 3. The powers of the equal director componentsscator are thus not revolving about the origin, but about the point 1

2 + 0 ex + 0 ey, as canbe seen from the above expression. If the scator coefficients have equal magnitude butopposite sign, an analogous scheme is obtained but in the plane at −45◦ with respect to theex, ey axes.

4.2. Unit magnitude scators with λ ratio of scator coefficients

In order to generalize to arbitrary hypercomplex director components, let ϕy = λϕx,where λ is an arbitrary real number. Any unit magnitude scator in S1+2 \

(S1+1

x ∪ S1+1y)

can be obtained with this scheme. From (26),

oϕ(1; ϕx, λϕx) = cos(ϕx) cos(λϕx) + cos(λϕx) sin(ϕx)ex + cos(ϕx) sin(λϕx)ey.

This function, with continuous arguments, also arises from the components exponentialmapping of line segments with constant scalar, inclined in the ex, ey plane [11]. The power

mappingoϕ −→ o

ϕm

may thus be viewed as a discrete version of the continuous componentsexponential mapping

ocexp

(t ex + λt ey

)= cos(t) cos(λt) + sin(t) cos(λt)e1 + cos(t) sin(λt)e2.

These curves no longer lie on a plane, as in the equal directors case, but a curve in threedimensional space lying on the cusphere surface. The curves are closed for rational λ. Forinteger λ, the function has periodicity π for λ odd; whereas there is a 2π period if λ is even.In the s, ex plane, the curves exhibit λ loops for odd λ and 2λ loops for even λ. For example,

the curve with λ = 3 is depicted in figure 4. Powers ofoϕ(1; π

8 , 3π8)

are shown in large dots,

whereas powers ofoϕ(1; π

32 , 3π32)

are shown in the smaller dots. Some other curves with λup to 4 are shown in Ref. [11]. These examples do not pretend to be exhaustive but to givea taste of the geometric representation of powers of scators.



17 of 22

Figure 4. Powers of scator, ϕy = λϕx with λ = 3. Powers ofoϕ(

1; π8 , 3π

8

)are shown in large

dots, whereas powers ofoϕ(

1; π32 , 3π

32

)are shown in the smaller dots. For example,

oϕ

2(1; π

8 , 3π8

)=

oϕ(

1; π4 , 3π

4

)is depicted by the large orange dot.

5. Exponential of a scator

The components exponential function mentioned in subsection 2.4, is defined in termsof products of exponentials with single component scator arguments. The question arisesas to whether the exponential of a scator, rather than the exponentials of its components, ispossible.

Lemma 4. The exponential of a scator z0 + zx ex + zyey ∈ S1+2 is

ez0+zx ex+zy ey = ez0 ezx ex ezy ey cosh(

zxzy

z0

)− ez0

(ezy ex ezx ey − cos

(zx − zy

))sinh

(zxzy

z0

). (29)

Proof. Commence with the exponential of a scator with only one director component,

ez0+zl el = ez0 ezl el = ez0(cos zl + sin zl el). (30a)

The above equalities are immediate from the general addition theorem and Euler’s rela-tionship for the imaginary unit el . However, it is necessary to recreate this result through alengthier procedure to establish two equations needed to obtain the exponential of a scatorin 1+2 dimensions. The transformation from multiplicative to additive (polar to Cartesian)variables in S1+1

l , from (5) is

z0 = ζ0 cos ζl , zl = ζ0 sin ζl el . (30b)

The series definition of the exponential function is

ez0+zl el =∞

∑m=0

(z0 + zl el)m

m!,



18 of 22

and in multiplicative variables

ez0+zl el = eζ0 cos ζl+ζ0 sin ζl el =∞

∑m=0

(ζ0 cos ζl + ζ0 sin ζl el)m

m!.

From the victoria equation in S1+nl , that in this 1+1 dimensional case, is identical to De

Moivre formula,

eζ0 cos ζl+ζ0 sin ζl el =∞

∑m=0

ζm0 cos(mζl)

m!+

∞

∑m=0

ζm0 sin(mζl)el

m!. (30c)

From the sum of (30c) and its conjugate

∞

∑m=0

ζm0 cos(mζl)

m!=

12

(eζ0 cos ζl+ζ0 sin ζl el + eζ0 cos ζl−ζ0 sin ζl el

).

Factor eζ0 cos ζl and use the cosine definition in terms of exponentials,

∞

∑m=0

ζm0 cos(mζl)

m!= eζ0 cos ζl cos(ζ0 sin ζl). (31)

The difference of (30c) minus its conjugate is,

∞

∑m=0

ζm0 sin(mζl)el

m!=

12

(eζ0 cos ζl+ζ0 sin ζl el − eζ0 cos ζl−ζ0 sin ζl el

).

An analogous procedure gives the series involving the sine function

∞

∑m=0

ζm0 sin(mζl)

m!= eζ0 cos ζl sin(ζ0 sin ζl). (32)

So thateζ0 cos ζl+ζ0 sin ζl el = eζ0 cos ζl cos(ζ0 sin ζl) + eζ0 cos ζl sin(ζ0 sin ζl)el .

Returning to the additive variables using (30b), (30a) is recovered.Consider the exponential of a scator with two director components in the additive

representation but with multiplicative variables

ez0+zx ex+zy ey = exp(ζ0 cos ζx cos ζy + ζ0 sin ζx cos ζyex + ζ0 sin ζy cos ζx ey

)=

∞

∑m=0

(ζ0 cos ζx cos ζy + ζ0 sin ζx cos ζyex + ζ0 sin ζy cos ζx ey

)m

m!,

from the multiplicative to additive transformation (7a),

z0 = ζ0 cos ζx cos ζy, zx = ζ0 sin ζx cos ζy, zy = ζ0 sin ζy cos ζx. (33)

From the victoria equation, a scator raised to the power m is equal to(ζ0 cos ζx cos ζy + ζ0 sin ζx cos ζyex + ζ0 sin ζy cos ζx ey

)m=

ζm0 cos(mζx) cos

(mζy

)+ ζm

0 sin(mζx) cos(mζy

)ex + ζm

0 sin(mζy

)cos(mζx)ey



19 of 22

so that

exp(ζ0 cos ζx cos ζy + ζ0 sin ζx cos ζyex + ζ0 sin ζy cos ζx ey

)=

∞

∑m=0

ζm0

cos(mζx) cos(mζy

)+ sin(mζx) cos

(mζy

)ex + sin

(mζy

)cos(mζx)ey

m!(34)

Let us evaluate each component in this expression.Scalar component. The product of cosine functions is rewritten as a sum of cosines,

cos(mζx) cos(mζy

)= cos

[m(ζx + ζy

)]+ ζm

0 cos[m(ζx − ζy

)], so that the series can be

evaluated from (31),

∞

∑m=0

ζm0 cos

[m(ζx ± ζy

)]m!

= eζ0 cos(ζx±ζy) cos[ζ0 sin

(ζx ± ζy

)].

The series of the product of cosine functions is then

∞

∑m=0

ζ0 cos(mζx) cos(mζy

)m!

=12

eζ0 cos(ζx+ζy) cos[ζ0 sin

(ζx + ζy

)]+

12

eζ0 cos(ζx−ζy) cos[ζ0 sin

(ζx − ζy

)]To recover the expression in terms of additive variables, expand the sum of angles and use(30b),

ζ0 cos(ζx ± ζy

)= ζ0 cos(ζx) cos

(ζy)∓ ζ0 sin(ζx) sin

(ζy)= z0 ∓

zxzy

z0(35a)

ζ0 sin(ζx ± ζy

)= ζ0 sin(ζx) cos

(ζy)± ζ0 sin

(ζy)

cos(ζx) = zx ± zy (35b)

to obtain,

∞

∑m=0


)m!

=12

ez0

(e−

zxzyz0 cos

(zx + zy

)+ e

zxzyz0 cos

(zx − zy

)).

The sum of angles in additive variables can again be expanded and regrouped in order torewrite the exponentials in terms of hyperbolic trigonometric functions

12

ez0

(e−

zxzyz0 cos

(zx + zy

)+ e

zxzyz0 cos

(zx − zy

))=

ez0 cosh(

zxzy

z0

)cos zx cos zy + ez0 sinh

(zxzy

z0

)sin zx sin zy.

The series of the product of cosine functions is then

∞

∑m=0


)m!

=

ez0 cosh(

zxzy

z0

)cos zx cos zy + ez0 sinh

(zxzy

z0

)sin zx sin zy (36a)

ex director component. Rewrite the product sin(mζx) cos(mζy

)ex in terms of the sum of sine

functions sin(mζx) cos(mζy

)= sin

[m(ζx + ζy

)]+ sin

[m(ζx − ζy

)]. Each series can then

be evaluated from (32),

∞

∑m=0

ζm0 sin

[m(ζx ± ζy

)]m!

= eζ0 cos(ζx±ζy) sin[ζ0 sin

(ζx ± ζy

)],



20 of 22

so that

∞

∑m=0

ζm0 sin(mζx) cos

(mζy

)m!

=12

eζ0 cos(ζx+ζy) sin[ζ0 sin

(ζx + ζy

)]+

12

eζ0 cos(ζx−ζy) sin[ζ0 sin

(ζx − ζy

)].

Expand the sum of angles from (35a) and (35b),

∞

∑m=0

ζm0 sin(mζx) cos

(mζy

)m!

=12

ez0−zxzy

z0 sin(zx + zy

)+

12

ez0+zxzy

z0 sin(zx − zy

).

Expand again the sum of angles and regroup to obtain

∞

∑m=0

ζm0 sin(mζx) cos

(mζy

)m!

=

ez0 cosh(

zxzy

z0

)sin zx cos zy − ez0 sinh

(zxzy

z0

)sin zy cos zx (36b)

ey director component. The product sin(mζy

)cos(mζx)ey is written in an analogous

fashion

∞

∑m=0

ζm0 sin

(mζy

)cos(mζx)

m!=

12

ez0−zxzy

z0 sin(zx + zy

)− 1

2ez0+

zxzyz0 sin

(zx − zy

).

In terms of hyperbolic functions

∞

∑m=0

ζm0 sin

(mζy

)cos(mζx)

m!=

ez0 cosh(

zxzy

z0

)sin zy cos zx − ez0 sinh

(zxzy

z0

)sin zx cos zy. (36c)

Collecting results from (36a), (36b) and (36c) in (34),

ez0+zx ex+zy ey = cosh(

zxzy

z0

)ez0(cos zx cos zy + sin zx cos zyex + sin zy cos zx ey

)+ sinh

(zxzy

z0

)ez0(sin zx sin zy − cos zx sin zyex − sin zx cos zyey

). (37)

Rewritten in terms of products of exponentials from [11, Eq.(16)], the exponential ofz0 + zx ex + zyey is

ez0+zx ex+zy ey = ez0 ezx ex ezy ey cosh(

zxzy

z0

)−(

ez0 ezy ex ezx ey − ez0 cos(zx − zy

))sinh

(zxzy

z0

).

The generalization to higher dimensions is carried out in a similar fashion, althoughthe algebra becomes considerably more elaborate.



21 of 22

The exponential of a scatoroϕ ∈ S1+2, in terms of the components exponential function

is

eoϕ = e f0+ fx ex+ fy ey =

ocexp

( oϕ)

cosh(

fx fy

f0

)−(o

cexp(

f0 + fyex + fx ey)− e f0 cos

(fx − fy

))sinh

(fx fy

f0

). (38)

Corollary 5. In the limit when the additive scalar component is much larger than the directorcomponents f 2

0 � f 2x , f 2

y , the exponential of a scator in S1+2 is equal to the components exponential

function of the scator, limf 20� f 2

x , f 2y

[exp

( oϕ)]

=ocexp

( oϕ).

Proof. If the argument is much smaller than one, the hyperbolic functions are,lim

| f0|�| fx fy|

[cosh

(fx fyf0

)]≈ 1 and lim

| f0|�| fx fy|

[sinh

(fx fyf0

)]≈ 0, from (38),

limf 20� f 2

x , f 2y

(e f0+ fx ex+ fy ey

)= e f0 e fx ex e fy ey =

ocexp

(f0 + fx ex + fyey

).

It is conjectured that this corollary should be true in 1+ n dimensions: The exponential

of a scatoroϕ ∈ S1+n is equal to the components exponential function of

oϕ in the limit

when the additive scalar component is much larger than any of the j director components,

limf 20� f 2

j

exp( o

ϕ)=

ocexp

( oϕ).

In this f 20 � f 2

x , f 2y limit, the scator magnitude becomes an Euclidean quadratic form,

since the productn∏j=1

(1 +

f 2j

f 20

)in (10) is then approximated by the sum,

limf 20� f 2

x , f 2y

ϕ20 = f 2

0

(n

∏j=1

(1 +

f 2j

f 20

))≈ f 2

0

(1 +

n

∑j=1

f 2j

f 20

)≈ f 2

0 +n

∑j=1

f 2j .

Geometrically, this limit corresponds to scators paraxial to the s axis. The intersection of the

cusphere and a constant s plane in this limit isn∑

j=1f 2j = ϕ2

0 − f 20 . This equation represents

an Sn−1 dimensional hypersphere embedded in an n dimensional space. For two director

components, S1 is a circle in the ex, ey plane with radius√

ϕ20 − f 2

0 .

In the limit when f0 → 0, if fx fy 6= 0, sinh(

fx fyf0

)≈ cosh

(fx fyf0

)≈ 1

2 exp(

fx fyf0

), the

exponential function

limf0→0

[exp

(f0 + fx ex + fyey

)]=

12

limf0→0

[cos(

fx − fy)

exp(

fx fy

f0

)]diverges as f0 → 0. This result is not unexpected because the scator product is defined inS1+2, where f0 cannot be zero if there are two non-vanishing director components. Theseries representation of the exponential with scator argument is

exp(

f0 + fx ex + fyey)=

∞

∑m=0

(f0 + fx ex + fyey

)m

m!,

this expression involves products of scators. For the second and third terms in the series,from Eq. (22) and (23) respectively, all three components of

(f0 + fx ex + fyey

)2 diverge



22 of 22

as f0 → 0. Therefore, the domain of the exponential function of scator argument is S1+2.However, the image of the exponential function is not restricted to the S1+2 set. Forexample,

exp(

f0 +π

2ex + πey

)= −e f0 cosh

(π2

2 f0

)ex + e f0 sinh

(π2

2 f0

)ey,

this scator has vanishing scalar but two non zero director components. The exponential

function mapping of a scator goes from the scator set onto the real set, exp( o

ϕ)

: S1+n →R1+n. In contrast, the components exponential function mapping is

ocexp

( oϕ)

: R1+n →S1+n.

Funding: This research received no external funding

Institutional Review Board Statement: Not applicable

Informed Consent Statement: Not applicable

Conflicts of Interest: The author declares no conflict of interest.

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